Standards for Mathematical Practice:
Standard 3: Construct Viable Arguments and Critique the Reasoning of Others
The Standard:
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in
constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.
They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their
conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making
plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a
flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects,
drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made
formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or
read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Classroom Observations:
Teachers who are developing students’ capacity to "construct viable arguments and critique the reasoning of others" require their
students to engage in active mathematical discourse. This might involve having students explain and discuss their thinking processes
aloud, or signaling agreement/disagreement with a hand signal. A middle childhood teacher might post multiple approaches to a
problem and ask students to identify plausible rationales for each approach as well as any mistakes made by the mathematician. An
early adolescence teacher might post a chart showing a cost-analysis comparison of multiple DVD rental plans and ask his students to
formulate and defend a way of showing when each plan becomes most economical. A teacher of adolescents and young adults might
actively engage her students in extended conjecture about conditions for proof in the construction of quadrilaterals, testing their
assumptions and questioning their approaches. Visit the video excerpts at Inside Mathematics website:
http://www.insidemathematics.org/index.php/mathematical-practice-standards to view multiple examples of teachers engaging students
in formulating, critiquing and defending arguments of mathematical reasoning.
Students:
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Make reasonable guesses to explore their ideas
Justify solutions and approaches
Listen to the reasoning of others, compare arguments, and
decide if the arguments of others makes sense
Ask clarifying and probing questions
Because Teachers:
 Provide opportunities for students to listen to or read the
conclusions and arguments of others
 Establish and facilitate a safe environment for discussion
 Ask clarifying and probing questions
 Avoid giving too much assistance (e.g., providing
answers or procedures)
Math Solutions
Math Practice
Construct viable
arguments and critique
the reasoning of others
Key Points
Students might think or do:
 make conjectures and build a logical progression of
statements to explore the truth of their conjectures
 analyze situations by breaking them into cases
 recognize and use counterexamples
 justify conclusions, communicate them to others, and
respond to the arguments of others
 distinguish correct logic or reasoning from that which is
flawed, and—if there is a flaw in an argument—explain
what it is
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Construct viable arguments and critique the reasoning of
others.
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Questions to Develop Mathematical Thinking
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Analyze problems and use stated mathematical assumptions,
definitions, and established results in constructing arguments.
Justify conclusions with mathematical ideas.
Listen to the arguments of others and ask useful questions to
determine if an argument makes sense.
Ask clarifying questions or suggest ideas to improve/revise the
argument.
Compare two arguments and determine correct or flawed logic.
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Practice
Construct
Viable
Arguments
and
Critique
Reasoning
of Others
Needs Improvement
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A student can state a rule for a pattern, and can
explain why their rule works for that pattern.
When someone claims “multiplying two numbers
gives you an answer bigger than either the
numbers,” a student can think about: - what happens
when you multiply 2 whole numbers; - what happens
when you multiply by a fraction; - what happens
when you multiply 2 fractions
Education Development Center, Inc.
What mathematical evidence would support your solution? How can we be
sure that...? / How could you prove that...? Will it still work if...?
What were you considering when...?
How did you decide to try that strategy?
How did you test whether your approach worked?
How did you decide what the problem was asking you to find? (What was
unknown?)
Did you try a method that did not work? Why didn’t it work? Would it ever
work? Why or why not?
What is the same and what is different about...?
How could you demonstrate a counter-example?
CCSS-M Flip Books: http://katm.org/wp/common-core/
Emerging
(teacher does thinking)
Task:
 Is ambiguously stated.
Task:
 Is not at the appropriate level.
Teacher:
 Does not ask students to
present arguments or solutions.
 Expects students to follow a
given solution path without
opportunities to make
conjectures.
Teacher:
 Does not help students differentiate
between assumptions and logical
conjectures.
 Asks students to present arguments but not
to evaluate them.
 Allows students to make conjectures without
justification.
Proficient
(teacher mostly models)
Exemplary
(students take ownership)
Task:
 Avoids single steps or routine
algorithms.
Teacher:
 Helps students differentiate
between assumptions and
logical conjectures.
 Prompts students to
evaluate peer arguments.
 Expects students to
formally justify the validity
of their conjectures.
Teacher:
 Identifies students’
assumptions.
 Models evaluation of student
arguments.
 Asks students to explain their
conjectures.
Institute for Advanced Study
Park City Mathematics Institute